Does that mean they have a case against me if I was asked to take a polygraph test?

Asking you to take a polygraph test does not show a strong case against you. Instead, it shows the opposite, because if they had a strong case they would just arrest you. The purpose of the request is to get you answering questions, and is supplemented by the "post-polygraph" questions they usually ask after they declare that you have failed the test. The police are allowed to lie to you to get you to make statements. You should not answer any police questions, period, with only three exceptions I can think of: 1. You are claiming self-defense. 2. You are claiming that you had permission from the owner to take the car, enter the home, etc. 3. You had a prescription for the drugs. The police know that polygraph tests are very unreliable, so unreliable that Michigan courts will not even allow them into evidence at a trial. You should tell the police that you will not be answering questions without your attorney present. If you have no attorney, then you will not be answering questions at all.